Q: A mechanic uses a wrench to loosen a bolt. The wrench is 0.30 m long, and the mechanic applies a force of 120 N at an angle of 45° to the handle. What is the magnitude of the torque produced?

$$T = Fr\sin\theta = (120\,\text{N})(0.30\,\text{m})\sin 45^\circ = 25.5\,\text{N-m}$$

Q: A 0.25 m wrench is used to tighten a bolt. If a torque of 40 N-m is required and the force is applied perpendicular to the wrench, what force must be exerted?

$$F = \frac{T}{r} = \frac{40\,\text{N-m}}{0.25\,\text{m}} = 160\,\text{N}$$

Q: A seesaw is 2.0 m long and massless. Child A (mass 40 kg) sits at one end. Where should Child B (mass 60 kg) sit to balance the seesaw?

$$r_B = \frac{m_A r_A}{m_B} = \frac{40\,\text{kg} \times 2.0\,\text{m}}{60\,\text{kg}} = 1.33\,\text{m}$$\text{ from the pivot}$$

Q: A solid wheel of mass 10 kg and radius 0.20 m is accelerated from rest to 10 rad/s in 5 s. What torque is required?

$$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{10\,\text{rad/s}}{5\,\text{s}} = 2\,\text{rad/s}^2$$ $$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 10 \times (0.20)^2$ = 0.2\,\text{kg}\cdot\text{m}^2$$ $$T = I \alpha = 0.2 \times 2 = 0.4\,\text{N-m}$$

Q: A thin hoop and a solid sphere both have mass $$m$$ and radius $$r. $$Which has the greater moment of inertia about its central axis?

The thin hoop, because $$I_{\text{hoop}} = m r^2$ and I$$_{\text{sphere}} = \frac{2}{5} m $$r^2$$

Q: A pulley of radius 0.15 m is pulled with a force of 12 N. If the rotational speed increases from 0 to 18 rev/min in 6 s, what is the moment of inertia of the pulley?

$$\omega_f = 18 \times \frac{2\pi}{60} = 1.885\,\text{rad/s}$$ $$\alpha = \frac{\omega_f}{t} = \frac{1.885}{6} = 0.314\,\text{rad/s}^2$$ $$T = F r = 12 \times 0.15 = 1.8\,\text{N-m}$$ $$I = \frac{T}{\alpha} = \frac{1.8}{0.314} = 5.73\,\text{kg}\cdot\text{m}^2$$

Q: A baton consists of a thin rod (negligible mass) and two masses of 0.40 kg at each end, separated by 0.80 m. What is the moment of inertia about the center?

$$I = 2 m r^2$ = 2 \times 0.40 \times (0.40)^2$ = 0.128\,\text{kg}\cdot\text{m}^2$$

Q: A cyclist of mass 70 kg stands on a pedal 0.18 m from the center of the chain ring, which is at an angle of 30° above the horizontal. What torque does the cyclist exert?

$$F = m g = 70 \times 9.8 = 686\,\text{N}$$ $$T = F r \sin\theta = 686 \times 0.18 \times \sin 30^\circ = 61.74\,\text{N-m}$$

Question 1

A mechanic uses a wrench to loosen a bolt. The wrench is 0.30 m long, and the mechanic applies a force of 120 N at an angle of 45° to the handle. What is the magnitude of the torque produced?

Accepted Answer

$$T = Fr\sin	heta = (120\,	ext{N})(0.30\,	ext{m})\sin 45^\circ = 25.5\,	ext{N-m}$$

Question 2

A door is 0.90 m wide. To open it with the least force, where and in what direction should you apply the force?

Accepted Answer

At the edge farthest from the hinges, perpendicular to the door

Question 3

A 0.25 m wrench is used to tighten a bolt. If a torque of 40 N-m is required and the force is applied perpendicular to the wrench, what force must be exerted?

Accepted Answer

$$F = \frac{T}{r} = \frac{40\,	ext{N-m}}{0.25\,	ext{m}} = 160\,	ext{N}$$

Question 4

A seesaw is 2.0 m long and massless. Child A (mass 40 kg) sits at one end. Where should Child B (mass 60 kg) sit to balance the seesaw?

Accepted Answer

$$r_B = \frac{m_A r_A}{m_B} = \frac{40\,	ext{kg} 	imes 2.0\,	ext{m}}{60\,	ext{kg}} = 1.33\,	ext{m}$$	ext{ from the pivot}$$

Question 5

A solid wheel of mass 10 kg and radius 0.20 m is accelerated from rest to 10 rad/s in 5 s. What torque is required?

Accepted Answer

$$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{10\,	ext{rad/s}}{5\,	ext{s}} = 2\,	ext{rad/s}^2$$ 
 $$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 10 	imes (0.20)^2$ = 0.2\,	ext{kg}\cdot	ext{m}^2$$ 
 $$T = I \alpha = 0.2 	imes 2 = 0.4\,	ext{N-m}$$

Question 6

Which of the following statements about moment of inertia is correct?

Accepted Answer

Moment of inertia depends on both the mass and its distribution relative to the axis of rotation.

Question 7

A thin hoop and a solid sphere both have mass $$m$$ and radius $$r. $$Which has the greater moment of inertia about its central axis?

Accepted Answer

The thin hoop, because $$I_{	ext{hoop}} = m r^2$ and I$$_{	ext{sphere}} = \frac{2}{5} m $$r^2$$

Question 8

A pulley of radius 0.15 m is pulled with a force of 12 N. If the rotational speed increases from 0 to 18 rev/min in 6 s, what is the moment of inertia of the pulley?

Accepted Answer

$$\omega_f = 18 	imes \frac{2\pi}{60} = 1.885\,	ext{rad/s}$$ 
 $$\alpha = \frac{\omega_f}{t} = \frac{1.885}{6} = 0.314\,	ext{rad/s}^2$$ 
 $$T = F r = 12 	imes 0.15 = 1.8\,	ext{N-m}$$ 
 $$I = \frac{T}{\alpha} = \frac{1.8}{0.314} = 5.73\,	ext{kg}\cdot	ext{m}^2$$

Question 9

A baton consists of a thin rod (negligible mass) and two masses of 0.40 kg at each end, separated by 0.80 m. What is the moment of inertia about the center?

Accepted Answer

$$I = 2 m r^2$ = 2 	imes 0.40 	imes (0.40)^2$ = 0.128\,	ext{kg}\cdot	ext{m}^2$$

Question 10

A cyclist of mass 70 kg stands on a pedal 0.18 m from the center of the chain ring, which is at an angle of 30° above the horizontal. What torque does the cyclist exert?

Accepted Answer

$$F = m g = 70 	imes 9.8 = 686\,	ext{N}$$ 
 $$T = F r \sin	heta = 686 	imes 0.18 	imes \sin 30^\circ = 61.74\,	ext{N-m}$$

Question 11

If two people pull on opposite sides of a wheel (diameter 2.0 m, mass 10 kg), one with 30 N clockwise and the other with 50 N counterclockwise, what is the net torque?

Accepted Answer

$$T_{	ext{net}} = (50 - 30) 	imes 1.0 = 20\,	ext{N-m}$$\,	ext{counterclockwise}

Question 12

A solid cylinder (mass 8 kg, radius 0.25 m) is accelerated by a torque of 2 N-m. What is its angular acceleration?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 8 	imes (0.25)^2$ = 0.25\,	ext{kg}\cdot	ext{m}^2$$ 
 $$\alpha = \frac{T}{I} = \frac{2}{0.25} = 8\,	ext{rad/s}^2$$

Question 13

A chain is wrapped around a pulley and pulled with a force of 16 N. The pulley has a radius of 0.20 m. What is the torque exerted?

Accepted Answer

$$T = F r = 16 	imes 0.20 = 3.2\,	ext{N-m}$$

Question 14

A solid wheel (mass 5 kg, radius 0.20 m) is at rest. You want it to rotate at 10 rev/s in 20 s. What force must you exert on a strap wrapped around the wheel?

Accepted Answer

$$\omega_f = 10 	imes 2\pi = 62.83\,	ext{rad/s}$$ 
 $$\alpha = \frac{\omega_f}{t} = \frac{62.83}{20} = 3.14\,	ext{rad/s}^2$$ 
 $$I = \frac{1}{2} m r^2$ = 0.5 	imes 5 	imes (0.20)^2$ = 0.1\,	ext{kg}\cdot	ext{m}^2$$ 
 $$T = I \alpha = 0.1 	imes 3.14 = 0.314\,	ext{N-m}$$ 
 $$F = \frac{T}{r} = \frac{0.314}{0.20} = 1.57\,	ext{N}$$

Question 15

A long, uniform rod of mass $$m$$ and length $$l is $$rotated about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12} m l^2$

Question 16

A solid sphere and a thin hoop both have mass 2 kg and radius 0.5 m. What is the ratio of their moments of inertia?

Accepted Answer

$$\frac{I_{	ext{hoop}}}{I_{	ext{sphere}}} = \frac{m r^2$}{\frac{2}{5} m r^2$} = \frac{5}{2} = 2.5$$

Question 17

A book is rotated about its edge and then about its center. Which requires more torque to achieve the same angular acceleration, and why?

Accepted Answer

Rotating about the edge requires more torque because the moment of inertia is greater.

Question 18

A solid wheel (mass 6 kg, radius 0.18 m) is accelerated by a force of 9 N applied at the rim. What is the angular acceleration?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = 0.5 	imes 6 	imes (0.18)^2$ = 0.0972\,	ext{kg}\cdot	ext{m}^2$$ 
 $$T = F r = 9 	imes 0.18 = 1.62\,	ext{N-m}$$ 
 $$\alpha = \frac{T}{I} = \frac{1.62}{0.0972} = 16.67\,	ext{rad/s}^2$$

Question 19

A ball rolls down a rough incline but slides down a smooth incline. Why does the ball roll on the rough surface?

Accepted Answer

Friction provides a torque that causes rotational motion, allowing the ball to roll.

Rotational Dynamics: Torque, Moment of Inertia, and Newton's Second Law for Rotation

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